Polynomial Graph Channels

• Polynomial Graph Channels are spectral graph filters that use polynomial approximations of graph operators (e.g., Laplacian, propagation matrix) for efficient signal processing.
• They adapt the polynomial basis—via learned orthonormal or signal-optimal techniques—to address diverse graph characteristics like homophily and heterophily.
• Recent approaches integrate piece-wise polynomial filtering and adaptive basis designs to enhance approximation accuracy while reducing computational complexity.

Polynomial graph channels are spectral graph filters—also referred to as graph channels—whose transfer functions are represented or approximated by polynomials of a graph operator such as the normalized Laplacian $L$, a scaled Laplacian $\widetilde L$, or a propagation matrix $P$. For a graph signal $x\in\mathbb R^n$, the general objective is to realize a filter $h$ without the $O(n^3)$ cost of full eigendecomposition by replacing $h$ with a degree-$K$ polynomial, while preserving the spectral selectivity needed for homophily, heterophily, or mixed regimes. Recent work places the polynomial basis itself at the center of model design: one line learns an orthonormal basis or computes a signal-optimal basis, another adapts the basis to graph heterophily, and a third partitions the spectrum and fits multiple local polynomials rather than one global polynomial (Guo et al., 2023, Huang et al., 2023, Lingam et al., 2021).

1. Spectral formulation of polynomial channels

Let $G=(V,E)$ be an undirected graph with normalized Laplacian

$L=I-D^{-1/2}AD^{-1/2},$

or equivalently a normalized adjacency or propagation operator such as $\widetilde L$0 or $\widetilde L$1. If

$\widetilde L$2

then a spectral filter acts on $\widetilde L$3 by

$\widetilde L$4

To avoid diagonalization, one approximates $\widetilde L$5 by a polynomial: $\widetilde L$6 where $\widetilde L$7 is a polynomial basis and $\widetilde L$8 are learnable coefficients. After scaling $\widetilde L$9 to $P$0, the same idea appears in Chebyshev form,

$P$1

and in the node domain as

$P$2

This establishes the basic equivalence between spectral polynomial filtering and repeated neighborhood propagation (Guo et al., 2023, Huang et al., 2023).

The polynomial basis is not merely a numerical convenience. A basis whose atoms $P$3 behave well on $P$4 for $P$5 or on $P$6 for $P$7 can improve numerical stability, localization, and convergence of gradient-based learning. Different classical bases impose different orthogonality relations, weight functions, and spectral biases. Monomials, Chebyshev, Bernstein, and Jacobi bases therefore induce materially different optimization and approximation behavior, even when the formal filter order $P$8 is the same (Guo et al., 2023).

2. Basis design as the central modeling choice

A recurring premise in this literature is that fixed, predefined polynomial bases are often too rigid. One formulation states that polynomial filters “typically use a predetermined polynomial basis and learn the coefficients from the training data,” while their effectiveness “is highly dependent on the property of the polynomial basis” (Guo et al., 2023). Another states that in the majority of polynomial filters the polynomials are “predefined and remain fixed across all graphs,” which fails to accommodate “the diverse heterophily degrees across different graphs” (Huang et al., 2023). A third observes that single-polynomial approaches may need to learn a reasonably high-order polynomial but can be ineffective at doing so because of their designs (Lingam et al., 2021).

Three complementary responses follow from these premises. The first is to learn the basis itself from the space of orthonormal polynomial systems. The second is to define an “optimal polynomial basis” for a given graph and graph signal, and then compute it tractably. The third is to abandon a single global polynomial and instead fit multiple adaptive polynomials over different spectral subsets. Taken together, these approaches treat basis selection as a structural inductive bias rather than a fixed preprocessing decision. This suggests that “polynomial graph channels” are best understood not as one architecture family, but as a design space organized around spectral parameterization, orthogonality, and graph-dependent basis adaptation.

3. Learnable and signal-optimal orthonormal bases

Classical orthonormal polynomial sequences satisfy a three-term recurrence. In the formulation used for graph filters,

$P$9

and for $x\in\mathbb R^n$0,

$x\in\mathbb R^n$1

with $x\in\mathbb R^n$2 and $x\in\mathbb R^n$3. Favard’s theorem gives the converse: any sequence satisfying such a recurrence is orthonormal for some positive weight $x\in\mathbb R^n$4. This theorem is used to construct FavardGNN, which parameterizes, for each layer and channel, recurrence coefficients $x\in\mathbb R^n$5 and recursively builds basis polynomials $x\in\mathbb R^n$6. With initialization $x\in\mathbb R^n$7, $x\in\mathbb R^n$8, the recurrence is

$x\in\mathbb R^n$9

The output on channel $h$0 is

$h$1

and the filtering routine applies $h$2 or $h$3 recursively without explicitly forming $h$4. By Favard’s theorem, this parameterization spans all possible orthonormal polynomial bases of degree $h$5, and $h$6 are learned jointly by back-propagation (Guo et al., 2023).

The same work develops OptBasisGNN by examining the “optimal polynomial basis” criterion of Wang & Zhang (2022). For the least-squares objective

$h$7

the Hessian is

$h$8

with $h$9 and $O(n^3)$0. The fastest convergence of gradient descent, with condition number $O(n^3)$1, occurs exactly when $O(n^3)$2, meaning that $O(n^3)$3 is orthonormal under weight $O(n^3)$4. Direct Gram–Schmidt against $O(n^3)$5 would require eigendecomposition of $O(n^3)$6 and cost $O(n^3)$7. OptBasisGNN instead constructs the basis implicitly through Lanczos-style orthonormalization of the Krylov sequence $O(n^3)$8, producing orthonormal vectors $O(n^3)$9 satisfying

$h$0

with filter output

$h$1

Each step requires one sparse-matrix multiply, two dot-products, and one scaling, for total complexity $h$2. The paper further states that OptBasisGNN is a special instance of FavardGNN where the learned $h$3 converge to those induced by the signal-dependent weight $h$4 (Guo et al., 2023).

4. Heterophily-adaptive universal bases

A distinct line of work relates polynomial basis design directly to graph heterophily. In this view, the filtered signal required on heterophilous graphs should emphasize higher frequencies. The central theoretical statement is a “Frequency Ratio” theorem: if $h$5 denotes the edge-homophily ratio and $h$6 is the optimal polynomial filter, then the Dirichlet energy

$h$7

More heterophilous graphs therefore require stronger high-frequency emphasis. A second result, the “Pivot-Angle” theorem, states that on a regular graph the expected frequency of a basis vector is a monotone increasing function of its angle from the constant signal. This motivates a basis construction that explicitly controls mutual angles among basis vectors (Huang et al., 2023).

The adaptive heterophily basis is built without labels. Its target angle is

$h$8

Starting from $h$9, the method generates a Krylov vector $K$0, orthonormalizes it by a three-term recurrence against the two previous vectors, forms a temporary average $K$1, computes a scalar $K$2, and defines

$K$3

The resulting vectors satisfy

$K$4

Thus the heterophily basis vectors are mutually equiangular at angle $K$5, which is presented as guaranteeing that they collectively span low through very high frequencies in a controlled way (Huang et al., 2023).

A pure heterophily basis can be suboptimal when $K$6, while the standard homophily basis $K$7 collapses asymptotically toward the dominant eigenvector on homophily graphs. UniBasis combines the two: $K$8 The resulting UniFilter is

$K$9

where $G=(V,E)$0 is learned end-to-end with supervised cross-entropy. In practice, the model fixes $G=(V,E)$1, chooses $G=(V,E)$2 near the graph’s estimated homophily $G=(V,E)$3, stacks the filter with an MLP head, and trains with Adam plus early stopping. Its complexity is $G=(V,E)$4, linear in graph size and $G=(V,E)$5 (Huang et al., 2023).

5. Piece-wise polynomial channels and spectral partitioning

Piece-wise polynomial filtering addresses a different limitation of single-channel polynomial filters: a single global polynomial may inadequately approximate filters with different behavior over different frequency ranges. PP-GNN begins with a normalized operator $G=(V,E)$6, where $G=(V,E)$7, and eigendecomposition

$G=(V,E)$8

The spectral interval is partitioned into disjoint contiguous bins

$G=(V,E)$9

and each bin defines a spectral projector

$L=I-D^{-1/2}AD^{-1/2},$0

On bin $L=I-D^{-1/2}AD^{-1/2},$1, a low-degree polynomial

$L=I-D^{-1/2}AD^{-1/2},$2

is learned, yielding the full filter

$L=I-D^{-1/2}AD^{-1/2},$3

This construction preserves polynomial efficiency inside each spectral region while allowing the global transfer function to be piece-wise adaptive (Lingam et al., 2021).

The learning pipeline first maps raw features $L=I-D^{-1/2}AD^{-1/2},$4 through a small MLP,

$L=I-D^{-1/2}AD^{-1/2},$5

then filters,

$L=I-D^{-1/2}AD^{-1/2},$6

and optimizes a node-wise cross-entropy objective. To enforce smooth transitions between adjacent spectral bins, PP-GNN adds boundary regularization,

$L=I-D^{-1/2}AD^{-1/2},$7

with total loss

$L=I-D^{-1/2}AD^{-1/2},$8

Two theoretical results organize the method. Theorem 4.1 states that approximating any target filter $L=I-D^{-1/2}AD^{-1/2},$9 with a piece-wise polynomial achieves no larger $\widetilde L$00 error than a single global $\widetilde L$01th-order polynomial. Theorem 4.2 states that if $\widetilde L$02 is the space of global degree-$\widetilde L$03 polynomials and $\widetilde L$04 augments it with adaptive degree-$\widetilde L$05 polynomials supported on top- and bottom-$\widetilde L$06 eigenvalues, then $\widetilde L$07 and

$\widetilde L$08

This is the formal expressive-gain argument for piece-wise polynomial channels (Lingam et al., 2021).

6. Empirical behavior, scalability, and recurring issues

Empirically, the three lines of work converge on the claim that basis adaptation matters most on heterophilous or spectrally mixed graphs. On node classification over heterophilous graphs such as Chameleon, Squirrel, and Actor, and citation networks such as Citeseer and Pubmed, FavardGNN and OptBasisGNN “consistently outperform fixed-basis methods” including GCN, ChebNet, BernNet, ChebNetII, and JacobiConv. An example reported for Chameleon is JacobiConv $\widetilde L$09 versus OptBasisGNN $\widetilde L$10. On large-scale benchmarks including ogbn-arxiv, ogbn-papers100M, and LINKX non-homophily datasets, OptBasisGNN scales via batch computation of $\widetilde L$11 and on ogbn-papers100M is reported as ChebNetII $\widetilde L$12 versus OptBasisGNN $\widetilde L$13. In synthetic multi-channel filter recovery on a YCbCr image dataset, OptBasis converges in $\widetilde L$14 epochs to $\widetilde L$15, while ChebNetII, Bernstein, and Favard take hundreds of epochs and Monomial never fully converges within $\widetilde L$16 epochs. The same study reports that FavardGNN, despite its expressiveness, can exhibit non-convex “bumps” and slow convergence, observed up to $\widetilde L$17 epochs, whereas OptBasisGNN avoids this by fixing the basis to the unique signal-optimal one (Guo et al., 2023).

UniFilter is evaluated on six citation, Wiki, and WebKB graphs, with Cora, Citeseer, and Pubmed described as $\widetilde L$18, and Actor, Chameleon, and Squirrel as $\widetilde L$19. Against “20+ strong baselines,” including ChebNet, GPR-GNN, BernNet, JacobiConv, OptBasisGNN, Specformer, GCN, GAT, $\widetilde L$20GCN, and GloGNN++, it “achieves the best or second-best accuracy on every dataset,” yields “especially large gains (1–6 % absolute) on strongly heterophilous graphs,” and in a synthetic homophily sweep remains “within 1 % of optimal across the entire $\widetilde L$21 range,” whereas HomFilter fails at low $\widetilde L$22, HetFilter fails at high $\widetilde L$23, and OrtFilter fluctuates. Spectrum-analysis further reports unimodal low-frequency emphasis on homophilous graphs and strong high-frequency weighting on heterophilous ones (Huang et al., 2023).

PP-GNN is evaluated on heterophilic datasets including Texas, Wisconsin, Cornell, Chameleon, Squirrel, and Flickr, and homophilic datasets including Cora, Citeseer, Pubmed, OGBN-ArXiv, Wiki-CS, Cora-Full, Computer, and Photos. It is reported to “win or tie on every dataset,” with examples including Squirrel $\widetilde L$24 versus GPR-GNN $\widetilde L$25, Chameleon $\widetilde L$26 versus $\widetilde L$27, and Texas/Wisconsin $\widetilde L$28 versus $\widetilde L$29. Homophilic gains are described as “more modest (1–2%),” but the method “never underperforms the best baseline.” Ablations show that homophilic graphs often need only $\widetilde L$30–$\widetilde L$31 eigenpairs, whereas heterophilic graphs benefit from $\widetilde L$32–$\widetilde L$33. For scalability, PP-GNN uses Lanczos-based routines to obtain top-$\widetilde L$34 and bottom-$\widetilde L$35 eigenpairs in $\widetilde L$36 time, and large graphs such as Flickr with $\widetilde L$37K nodes and $\widetilde L$38K edges and OGBN-ArXiv with $\widetilde L$39K nodes and $\widetilde L$40M edges are handled using $\widetilde L$41 eigenpairs. End-to-end training is reported as only $\widetilde L$42 slower than GPR-GNN (Lingam et al., 2021).

Across these results, a common misconception is that polynomial channels differ mainly in coefficient parameterization while the basis is secondary. The cited work instead makes basis choice the primary object of theory and algorithm design. Another common assumption is that “optimal” or highly adaptive bases necessarily require eigendecomposition; OptBasisGNN is explicitly presented as a counterexample with $\widetilde L$43 construction, while PP-GNN accepts partial eigendecomposition in exchange for greater spectral flexibility. A plausible implication is that future polynomial graph channels will continue to trade off four axes already visible in these models: global versus piece-wise approximation, fixed versus learned orthogonality, signal-optimal versus graph-level adaptive bases, and purely sparse propagation versus selective spectral computation.